Which section goes first

Zero in, and not zero out puts the arithmetic’s own rounding into a cascade of biquads and measures what it adds. It takes the sections in whatever order the root finder produced them, and says so in passing. This essay asks what that choice was worth.

It is the third item on the same filter, rounded twice’s list of what it did not do:

Section ordering and scaling in a cascade decide overflow and internal round-off noise, which are distinct from the pole positions the essay measures. Not covered.

Four sections can be arranged in twenty-four orders. All twenty-four have the same poles, the same zeros and the same transfer function to the last digit — the cascade is a product and multiplication commutes. What differs is everything that is not the transfer function.

Two costs, one run #

An ordering is paid for at both ends of the word.

At the top, a section can overflow. Each section’s output is a signal in a fixed-point word with a fixed range, and a resonant section multiplies whatever reaches it by its own peak gain. The quantity that matters is the largest value any section’s output reaches for a full-scale input, and it is measured on the exact march rather than the rounded one — a rounded march that has already overflowed is measuring the overflow.

At the bottom, a section adds rounding noise. The rung below establishes where that noise goes: it is injected at a section’s output, circulates through that section’s own feedback, and is then filtered by every section after it. A resonant section injects a lot, because its own recursion amplifies whatever it is given; whether that reaches the output depends on what follows it.

The input is a seeded broadband sequence rather than a tone, for two reasons. A single tone below the corner excites none of the sections near their peaks, so neither cost appears. And a seeded sequence means every one of the twenty-four orderings sees the same input to the bit, which is what makes twenty-four numbers comparable rather than twenty-four experiments.

Twenty-four points #

The floor runs from 24.2 to 99.8 least significant bits — a factor of 4.1 — and the largest internal value from 0.088 to 0.699 of full scale, which is 18.0 decibels of headroom. Neither is a small effect and neither is visible in the response.

The two are not the same ordering. The quietest arrangement peaks at 0.699, which is the worst headroom of all twenty-four. The arrangement with the most headroom has a floor of 56 least significant bits, which is well over twice the best.

Counting properly: thirteen of the twenty-four orderings are beaten on both counts by some other ordering, and are simply mistakes — there is no reason to choose them and no specification they win on. The remaining eleven sit on a frontier and are actual choices. A design that picks an ordering because it is the order the design tool emitted has better than an even chance of picking a mistake.

The rule of thumb, and where it comes from #

There is a rule of thumb in the literature and it says to put the section with the highest quality factor last for one purpose and first for another, which is the sort of advice that gets remembered backwards. Both halves are measurable, and averaging over all six orderings that put the most resonant section in each position removes everything else.

Moving the most resonant section from first to last takes the round-off floor from 31.3 to 58.9 least significant bits and the largest internal value from 0.479 to 0.144 — a factor of 3.3, or 10.5 decibels of headroom bought at the price of six decibels of noise floor.

The mechanism for each half is one sentence.

Noise wants it first. The noise a resonant section injects is large, and everything after it is a lowpass, so putting it early means three sections of filtering stand between its noise and the output. Putting it last means that noise reaches the output untouched.

Headroom wants it last. A resonant section multiplies its input by its own peak gain. Put it first and it is handed the full-scale input; put it last and it is handed something three lowpass sections have already attenuated.

Both are consequences of the same asymmetry — that a cascade is ordered even though its product is not — and they point in opposite directions because one of them is about what a section receives and the other is about what it emits.

How much the trade is worth depends on the family #

An eighth-order Butterworth’s four sections have pole radii of 0.9767, 0.9713, 0.9663 and 0.9629 — close together, because a Butterworth’s poles are equally spaced on a circle. Its floor runs from 5.9 to 12.2 least significant bits and its peak over 5.0 decibels, and fifteen of its twenty-four orderings are on the frontier because so many of them are nearly equivalent.

The Chebyshev’s radii are 0.9945, 0.9843, 0.9766 and 0.9724, and the spread of the first one from the rest is what does it: one section with a radius three times closer to the unit circle than the others dominates both costs, and where it sits is nearly the whole answer.

So the ordering question is a question about how unequal the sections are, and three families, one corner is where that inequality is chosen. A designer who takes an elliptic response for its skirt is also taking a cascade whose sections differ by more, and therefore an ordering decision that matters more.

What this does not change, and what it does #

None of it changes the transfer function. Every one of the twenty-four orderings has the same magnitude, the same phase and the same group delay, exactly, and a frequency-response measurement of any of them is a measurement of all of them.

That is worth dwelling on because it makes the whole quantity invisible to the obvious instrument. The corner that moved is a defect a swept measurement finds immediately; this one it cannot find at all, because there is nothing wrong with the response. What finds it is a full-scale transient — for the overflow — and a silent input, for the floor.

And a factor of four in noise is two bits. A designer choosing between a sixteen-bit and an eighteen-bit machine, at some cost, may be choosing between two orderings of the machine they already have.

Which section’s noise it actually is #

The noise budget can be read section by section, and doing so says something the two curves do not.

In the quietest ordering — the most resonant section first — the noise budget is 88.7 per cent from the last section, which is the least resonant of the four. Its own recursion is mild and its contribution is small, and it is nearly the whole of the total only because everything before it has been filtered by three lowpass sections on the way out.

In the noisiest ordering the largest single contribution is 45.9 per cent, from the second section — which is neither the first nor the resonant one. That noise is injected in the middle of the chain and then passes through the resonant section, which multiplies it at its own resonance.

So the rule is not “the resonant section is noisy”. It is that the resonant section amplifies whatever is upstream of it, and putting it last puts three sections upstream. That is why the trend in position is monotone rather than a step, and it is why the effect is a property of the whole chain rather than of one part.

Where the measurement itself runs out #

The ratio between the best and worst orderings is 4.1 at sixteen bits, 3.5 at eighteen and 3.6 at twenty — stable, as it should be, because the whole quantity is measured in least significant bits and those scale with the word.

Below sixteen it is not. At fourteen bits the ratio reads 7.0 and at twelve it reads 4.4, and the reason is the rung below’s own boundary: below about twelve bits the rounding stops being a white sequence and the dead band contributes an offset of its own, so what is being compared is no longer two noise floors. Zero in, and not zero out puts that boundary between eleven and thirteen bits for this filter, and it arrives here as a limit on the comparison rather than on the filter.

That is worth stating because it is easy to read the wrong way round. The ordering matters more at short word lengths, not less; what stops being reliable is the measurement of by how much.

What scaling is, and why it is not settled here #

Ordering is half of the omission the rung below recorded; scaling is the other half and is only partly addressed.

Every section here is scaled to unity gain at direct current, which is one defensible choice and is what makes the cascade’s overall gain one without a correction factor at the end. The alternatives scale each section so that its output has a stated peak — for a stated input, in a stated norm — and they trade the same two quantities differently: an L-infinity scaling guarantees no overflow for any bounded input and costs noise, and an L-two scaling is quieter and can overflow on a signal that is unusual rather than large.

The measurement above holds the scaling fixed and varies the order, which isolates one of the two. Varying both is a two-dimensional search with a real literature behind it, and the honest statement is that this essay measures the axis that is discrete and leaves the one that is continuous.

Twenty-four is small, and eight sections is not #

Four sections have twenty-four orderings and an exhaustive answer costs a fraction of a second. Five have a hundred and twenty, six have seven hundred and twenty, and an order-sixteen filter’s eight sections have forty thousand — still enumerable, at a few minutes, and the point at which somebody reaches for a heuristic instead.

The measurement above says what a heuristic should be. Both trends are monotone in the position of the most resonant section, and the effect is dominated by how unequal the sections are, so the ordering by pole radius is a good ordering and the two ends of it are the two ends of the trade. A rule that says “sort by radius, and choose which end from whether the design is short of headroom or short of floor” recovers most of what an enumeration finds.

What it does not recover is the count of mistakes. Thirteen of twenty-four here are dominated, and the proportion is a property of the family rather than of the rule — nine of twenty-four for the Butterworth, because its sections are nearly interchangeable. A heuristic tells a designer what to do; only the enumeration tells them how much of the space they would have been choosing from at random.

What is not in this model #

No overflow, still. The peak is measured and nothing is done about it: the arithmetic here does not wrap or saturate, so a value above full scale is simply a large number. The rung below records the same gap, and it matters more here, because the whole point of measuring the peak is what happens when it is exceeded — and a wrap-around overflow in a recursion produces a large limit cycle rather than a small distortion.

No direct form for comparison. The cascade is compared with itself. A direct form has no ordering at all and one quantiser, and both of its costs are worse than any of these twenty-four; that is the rung below’s finding for the coefficients and is not re-measured here for the arithmetic.

And no ladder. A ladder is not a cascade is the analogue field’s version of the same statement, and the digital equivalents — lattice and wave-digital structures — have neither an ordering problem nor a coefficient-sensitivity one, at the cost of more arithmetic per sample. They are the right answer to this whole essay and are not built here.

What a measurement of this looks like #

Both halves are measurable on hardware and neither is measurable with a swept sine.

The overflow half is found with a full-scale step or a burst at the filter’s own corner, watching an internal node rather than the output. On a signal processor that means reading the accumulator or the section state, which is a debugger’s job rather than an instrument’s, and it is why the failure is usually found as an audible click on a transient rather than as a number.

The noise half is found with the input tied to zero or to a dithered least significant bit, and measured on the output’s spectrum with everything else quiet. Four decibels of difference between two orderings is easy to see and impossible to attribute without knowing what to look for, because the response is identical: two builds of the same filter, one four times noisier, with nothing in the frequency response to distinguish them.

The floor a converter sets is the floor this one has to be compared against, and that is the practical test of whether the ordering matters at all. A cascade whose worst ordering sits below the converter’s own floor has no ordering problem; the Chebyshev above at sixteen bits does not, because a hundred least significant bits of its own arithmetic is far above anything a sixteen-bit converter contributes.

The habit this belongs to #

Twenty-four is a small enough number to enumerate, and enumerating it turned a rule of thumb into two monotone curves and a count of how many arrangements are strictly wrong. Neither curve was in doubt qualitatively — everyone who has built one of these knows that the high-Q section wants to be late and the headroom wants it early — and what was missing was the exchange rate between them, which is the only thing that decides whether the choice is worth making on a particular part.

That is the same move the three tolerances that do nothing makes with thirty-two corners of a tolerance box: where the set of possibilities is finite and small, the honest thing is to look at all of it rather than to sample it or to quote a rule about it. The rule of thumb turned out to be right, in both directions, and neither half of it says by how much — and by how much is the whole of whether the choice is worth making.

It is also worth noticing what enumeration is not available for, because the boundary is close. Four sections give twenty-four orderings and six give seven hundred and twenty; the same question asked about which pole pair is paired with which zero pair multiplies that again, and the design space stops being enumerable somewhere around order twelve. What the sibling essay in the networks field does at that point is instructive: the three tolerances that do nothing abandons the six hundred samples entirely and computes the whole second-derivative matrix, which says outright what no sample could — that two of five eigenvalues are of order one and the other three are nine decades down, a three-dimensional subspace a sample of a five-dimensional box never lands on. The equivalent move here would be a derivative of the round-off floor with respect to the section ordering, and there is no such thing, because an ordering is not a continuous parameter.

So this measurement is enumerable and not differentiable, and the tolerance measurements are differentiable and not enumerable. Both are exhaustive in the sense that matters and neither method transfers, which is worth stating because the two essays otherwise read as the same argument.